Yes, I think the instability comes from this design not the OPAmp itself. E-design, your last figure doesn't show the loop gain, the DC gain of the open-loop system should be around 100dB for this system. So the phase margin is not that what you show, and probably your simulation is wrong.
1 Opamp is the best, most straightforward solution, I also recommend that.
It is nice you changed R2 to 75kOhm. I tried in Tina the circuit when I changed only the R2 to 75kOhm and the PM changed from -23° to +28°. I added the 2.2nF and the PM increased really to 108° by an LHP zero, it is too much a bit.
This is expectable. With the bjt (0.6V thresold voltage) you need an Opamp which output goes close to its positive rail.y 24,34V and no input (0V) I have exactly 23V on the base of the transistor which leads to ~7mA current through the load
I suggest to check also the current source output resistance in simulation. It reveals about 50k output resistance. Unfortunately the provided "20 mA trim" resistor combination doesn't coincide with high output impedance.
See attached a calculation of OP current source dimensioning for infinite output resistance.
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